
theorem Th17:
  for X be non empty set, r be negative Real, f be Function of X,ExtREAL
    holds f is without+infty iff r(#)f is without-infty
proof
   let X be non empty set, r be negative Real, f be Function of X,ExtREAL;
   thus f is without+infty implies r(#)f is without-infty;
   assume A2: r(#)f is without-infty;
   now let x be set;
    assume x in dom f; then
A3: x in dom(r(#)f) by MESFUNC1:def 6; then
    (r(#)f).x = r * f.x by MESFUNC1:def 6; then
    r * f.x > -infty by A2,A3,MESFUNC5:10; then
    f.x <> +infty by XXREAL_3:def 5;
    hence f.x < +infty by XXREAL_0:4;
   end;
   hence f is without+infty by MESFUNC5:11;
end;
